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Preface Chapter 1. Introduction 1. Preliminaries and notation 2. Partial differential equations Additional material: More on normed vector spaces and metric spaces Problems Chapter 2. Where do PDE come from 1. An example: Maxwell's equations 2. Euler-Lagrange equations Problems Chapter 3. First order scalar semilinear equations Additional material: More on ODE and the inverse function theorem Problems Chapter 4. First order scalar quasilinear equations Problems Chapter 5. Distributions and weak derivatives Additional material: The space I Problems Chapter 6. Second order constant coefficient PDE: Types and d'Alembert's solution of the wave equation S1. Classification of second order PDE S2. Solving second order hyperbolic PDE on R2 Problems Chapter 7. Properties of solutions of second order PDE: Propagation, energy estimates and the maximum principle 1. Properties of solutions of the wave equation: Propagation phenomena 2. Energy conservation for the wave equation 3. The maximum principle for Laplace's equation and the heat equation 4. Energy for Laplace's equation and the heat equation Problems Chapter 8. The Fourier transform:Basic properties,the inversion formula and the heat equation 1. The definition and the basics 2. The inversion formula 3. The heat equation and convolutions 4. Systems of PDE 5.Integral transforms Additional material: A heat kernel proof of the Fourier inversion formula Problems Chapter 9. The Fourier transform:Tempered distributions,the wave equation and Laplace's equation 1. Tempered distributions 2. The Fourier transform of tempered distributions 3. The wave equation and the Fourier transform 4. More on tempered distributions Problems Chapter 10. PDE and boundaries 1. The wave equation on a half space 2. The heat equation on a half space 3. More complex geometries 4. Boundaries and properties of solutions 5. PDE on intervals and cubes Problems Chapter 11. Duhamel's principle 1. The inhomogeneous heat equation 2. The inhomogeneous wave equation Problems Chapter 12. Separation of variables 1. The general method 2. Interval geometries 3. Circular geometries Problems Chapter 13. Inner product spaces, sy mmetric operators, orthogonality 1. The basics of inner product spaces 2. Symmetric operators 3. Completeness of orthogonal sets and of the inner product space Problems Chapter 14. Convergence of the Fourier series and the Poisson formula on disks 1. Notions of convergence 2. Uniform convergence of the Fourier transform 3. What does the Fourier series converge to 4. The Dirichlet problem on the disk Additional material: The Dirichlet kernel Problems Chapter 15. Bessel functions 1. The definition of Bessel functions 2. The zeros of Bessel functions 3. Higher dimensions Problems Chapter 16. The method of stationary phase Problems Chapter 17. Solvability via duality 1. The general method 2. An example: Laplace's equation 3. Inner product spaces and solvability Problems Chapter 18. Variational problems 1. The finite dimensional problem 2. The infinite dimensional minimization Problems Bibliography Index